Abstract Group Automorphisms of Heisenberg Groups and Partial Automatic Continuity

Tomoya Tatsuno

Published 2024-01-28Version 1

Any abstract (not necessarily continuous) group automorphism of a simple, compact Lie group must be continuous due to Cartan (1930) and van der Waerden (1933). The purpose of this paper is to study a similar question in nilpotent Lie groups. Let $N$ be the 2-step Iwasawa N-group of a simple Lie group of rank 1. The group $N$ is precisely one of the $(2n+1)$-dimensional Heisenberg group, $(4n+3)$-dimensional quaternionic Heisenberg group, and 15-dimensional octonionic Heisenberg group. We show that any abstract (not necessarily continuous) group automorphism of $N$ is a product of a (possibly discontinuous) central automorphism and a Lie group automorphism. Thus, the discontinuity only occurs at the center for these 2-step nilpotent Lie groups. Moreover, we gain a similar result about abstract homomorphisms under certain conditions. We give a uniform proof from a geometric perspective. We also present the first example of a nilpotent Lie group whose automorphism group is not of the type described above.